11.1Band Theory of Solids

1
CHAPTER 11Semiconductor Theory and Devices

• 11.1 Band Theory of Solids
• 11.2 Semiconductor Theory
• 11.3 Semiconductor Devices
• 11.4 Nanotechnology

It is evident that many years of research by a
great many people, both before and after the
discovery of the transistor effect, has been
required to bring our knowledge of semiconductors
to its present development. We were fortunate to
be involved at a particularly opportune time and
to add another small step in the control of
Nature for the benefit of mankind. - John
Bardeen, 1956 Nobel lecture
2
11.1 Band Theory of Solids

• In Chapter 10 you learned about structural,
  thermal, and magnetic properties of solids.
• In this chapter we concentrate on electrical
  conduction.

3
Categories of Solids

• There are three categories of solids, based on
  their conducting properties
• conductors
• semiconductors
• insulators

4
Electrical Resistivity and Conductivity of
Selected Materials at 293 K
0
5
Reviewing the previous table reveals that

• The electrical conductivity at room temperature
  is quite different for each of these three kinds
  of solids
• Metals and alloys have the highest conductivities
  
• followed by semiconductors
• and then by insulators

6
Semiconductor Conduction

• The free-electron model from Chapter 9 does not
  apply to semiconductors and insulators, since
  these materials simply lack enough free electrons
  to conduct in a free-electron mode.
• There is a different conduction mechanism for
  semiconductors than for normal conductors.

7
Resistivity vs. Temperature
Figure 11.1 (a) Resistivity versus temperature
for a typical conductor. Notice the linear rise
in resistivity with increasing temperature at all
but very low temperatures. (b) Resistivity versus
temperature for a typical conductor at very low
temperatures. Notice that the curve flattens and
approaches a nonzero resistance as T ? 0. (c)
Resistivity versus temperature for a typical
semiconductor. The resistivity increases
dramatically as T ? 0.
8
Band Theory of Solids

• In order to account for decreasing resistivity
  with increasing temperature as well as other
  properties of semiconductors, a new theory known
  as the band theory is introduced.
• The essential feature of the band theory is that
  the allowed energy states for electrons are
  nearly continuous over certain ranges, called
  energy bands, with forbidden energy gaps between
  the bands.

9
Band Theory of Solids

• Consider initially the known wave functions of
  two hydrogen atoms far enough apart so that they
  do not interact.

10
Band Theory of Solids

• Interaction of the wave functions occurs as the
  atoms get closer
• An atom in the symmetric state has a nonzero
  probability of being halfway between the two
  atoms, while an electron in the antisymmetric
  state has a zero probability of being at that
  location.

Symmetric
Antisymmetric
11
Band Theory of Solids

• In the symmetric case the binding energy is
  slightly stronger resulting in a lower energy
  state.
• Thus there is a splitting of all possible energy
  levels (1s, 2s, and so on).
• When more atoms are added (as in a real solid),
  there is a further splitting of energy levels.
  With a large number of atoms, the levels are
  split into nearly continuous energy bands, with
  each band consisting of a number of closely
  spaced energy levels.

12
Kronig-Penney Model

• An effective way to understand the energy gap in
  semiconductors is to model the interaction
  between the electrons and the lattice of atoms.
• R. de L. Kronig and W. G. Penney developed a
  useful one-dimensional model of the electron
  lattice interaction in 1931.

13
Kronig-Penney Model

• Kronig and Penney assumed that an electron
  experiences an infinite one-dimensional array of
  finite potential wells.
• Each potential well models attraction to an atom
  in the lattice, so the size of the wells must
  correspond roughly to the lattice spacing.

14
Kronig-Penney Model

• Since the electrons are not free their energies
  are less than the height V0 of each of the
  potentials, but the electron is essentially free
  in the gap 0 lt x lt a, where it has a wave
  function of the form
• where the wave number k is given by the usual
  relation

15
Tunneling

• In the region between a lt x lt a b the electron
  can tunnel through and the wave function loses
  its oscillatory solution and becomes exponential

16
Kronig-Penney Model

• Matching solutions at the boundary, Kronig and
  Penney find
• Here K is another wave number.

17
Kronig-Penney Model

• The left-hand side is limited to values between
  1 and -1 for all values of K.
• Plotting this it is observed there exist
  restricted (shaded) forbidden zones for solutions.

18
The Forbidden Zones

• Figure 11.5 (a) Plot of the left side of Equation
  (11.3) versus ka for ?2ba / 2 3p / 2. Allowed
  energy values must correspond to the values of k
  for
• for which the plotted function lies
  between -1 and 1. Forbidden values are shaded in
  light blue. (b) The corresponding plot of energy
  versus ka for ?2ba / 2 3p / 2, showing the
  forbidden energy zones (gaps).

19
Important differences between the Kronig-Penney
model and the single potential well

1. For an infinite lattice the allowed energies
   within each band are continuous rather than
   discrete. In a real crystal the lattice is not
   infinite, but even if chains are thousands of
   atoms long, the allowed energies are nearly
   continuous.
2. In a real three-dimensional crystal it is
   appropriate to speak of a wave vector . The
   allowed ranges for constitute what are referred
   to in solid state theory as Brillouin zones.

20
And

• In a real crystal the potential function is more
  complicated than the Kronig-Penney squares. Thus,
  the energy gaps are by no means uniform in size.
  The gap sizes may be changed by the introduction
  of impurities or imperfections of the lattice.
• These facts concerning the energy gaps are of
  paramount importance in understanding the
  electronic behavior of semiconductors.

21
Band Theory and Conductivity

• Band theory helps us understand what makes a
  conductor, insulator, or semiconductor.
• Good conductors like copper can be understood
  using the free electron
• It is also possible to make a conductor using a
  material with its highest band filled, in which
  case no electron in that band can be considered
  free.
• If this filled band overlaps with the next higher
  band, however (so that effectively there is no
  gap between these two bands) then an applied
  electric field can make an electron from the
  filled band jump to the higher level.
• This allows conduction to take place, although
  typically with slightly higher resistance than in
  normal metals. Such materials are known as
  semimetals.

22
Valence and Conduction Bands

• The band structures of insulators and
  semiconductors resemble each other qualitatively.
  Normally there exists in both insulators and
  semiconductors a filled energy band (referred to
  as the valence band) separated from the next
  higher band (referred to as the conduction band)
  by an energy gap.
• If this gap is at least several electron volts,
  the material is an insulator. It is too difficult
  for an applied field to overcome that large an
  energy gap, and thermal excitations lack the
  energy to promote sufficient numbers of electrons
  to the conduction band.

23
Smaller energy gaps create semiconductors

• For energy gaps smaller than about 1 electron
  volt, it is possible for enough electrons to be
  excited thermally into the conduction band, so
  that an applied electric field can produce a
  modest current.
• The result is a semiconductor.

24
11.2 Semiconductor Theory

• At T 0 we expect all of the atoms in a solid to
  be in the ground state. The distribution of
  electrons (fermions) at the various energy levels
  is governed by the Fermi-Dirac distribution of
  Equation (9.34)
• ß (kT)-1 and EF is the Fermi energy.

25
Temperature and Resistivity

• When the temperature is increased from T 0,
  more and more atoms are found in excited states.
• The increased number of electrons in excited
  states explains the temperature dependence of the
  resistivity of semiconductors.
• Only those electrons that have jumped from the
  valence band to the conduction band are available
  to participate in the conduction process in a
  semiconductor. More and more electrons are found
  in the conduction band as the temperature is
  increased, and the resistivity of the
  semiconductor therefore decreases.

26
Some Observations

• Although it is not possible to use the
  Fermi-Dirac factor to derive an exact expression
  for the resistivity of a semiconductor as a
  function of temperature, some observations
  follow
• The energy E in the exponential factor makes it
  clear why the band gap is so crucial. An increase
  in the band gap by a factor of 10 (say from 1 eV
  to 10 eV) will, for a given temperature, increase
  the value of exp(ßE) by a factor of exp(9ßE).
• This generally makes the factor FFD so small
  that the material has to be an insulator.
• Based on this analysis, the resistance of a
  semiconductor is expected to decrease
  exponentially with increasing temperature.
• This is approximately truealthough not exactly,
  because the function FFD is not a simple
  exponential, and because the band gap does vary
  somewhat with temperature.

27
Clement-Quinnell Equation

• A useful empirical expression developed by
  Clement and Quinnell for the temperature
  variation of standard carbon resistors is given
  by
• where A, B, and K are constants.

28
Test of the Clement-Quinnell Equation
Figure 11.7 (a) An experimental test of the
Clement-Quinnell equation, using resistance
versus temperature data for four standard carbon
resistors. The fit is quite good up to 1 / T
0.6, corresponding to T 1.6 K. (b) Resistance
versus temperature curves for some thermometers
used in research. A-B is an Allen-Bradley carbon
resistor of the type used to produce the curves
in (a). Speer is a carbon resistor, and CG is a
carbon-glass resistor. Ge 100 and 1000 are
germanium resistors. From G. White, Experimental
Techniques in Low Temperature Physics, Oxford
Oxford University Press (1979).
29
Holes and Intrinsic Semiconductors

• When electrons move into the conduction band,
  they leave behind vacancies in the valence band.
  These vacancies are called holes. Because holes
  represent the absence of negative charges, it is
  useful to think of them as positive charges.
• Whereas the electrons move in a direction
  opposite to the applied electric field, the holes
  move in the direction of the electric field.
• A semiconductor in which there is a balance
  between the number of electrons in the conduction
  band and the number of holes in the valence band
  is called an intrinsic semiconductor.
• Examples of intrinsic semiconductors include
  pure carbon and germanium.

30
Impurity Semiconductor

• It is possible to fine-tune a semiconductors
  properties by adding a small amount of another
  material, called a dopant, to the semiconductor
  creating what is a called an impurity
  semiconductor.
• As an example, silicon has four electrons in its
  outermost shell (this corresponds to the valence
  band) and arsenic has five.
• Thus while four of arsenics outer-shell
  electrons participate in covalent bonding with
  its nearest neighbors (just as another silicon
  atom would), the fifth electron is very weakly
  bound.
• It takes only about 0.05 eV to move this extra
  electron into the conduction band.
• The effect is that adding only a small amount of
  arsenic to silicon greatly increases the
  electrical conductivity.

31
n-type Semiconductor

• The addition of arsenic to silicon creates what
  is known as an n-type semiconductor (n for
  negative), because it is the electrons close to
  the conduction band that will eventually carry
  electrical current.
• The new arsenic energy levels just below the
  conduction band are called donor levels because
  an electron there is easily donated to the
  conduction band.

32
Acceptor Levels

• Consider what happens when indium is added to
  silicon.
• Indium has one less electron in its outer shell
  than silicon. The result is one extra hole per
  indium atom. The existence of these holes creates
  extra energy levels just above the valence band,
  because it takes relatively little energy to move
  another electron into a hole
• Those new indium levels are called acceptor
  levels because they can easily accept an electron
  from the valence band. Again, the result is an
  increased flow of current (or, equivalently,
  lower electrical resistance) as the electrons
  move to fill holes under an applied electric
  field
• It is always easier to think in terms of the flow
  of positive charges (holes) in the direction of
  the applied field, so we call this a p-type
  semiconductor (p for positive).
• acceptor levels p-Type semiconductors
• In addition to intrinsic and impurity
  semiconductors, there are many compound
  semiconductors, which consist of equal numbers of
  two kinds of atoms.

33
Thermoelectric Effect

• In one dimension the induced electric field E in
  a semiconductor is proportional to the
  temperature gradient, so that
• where Q is called the thermoelectric power.
• The direction of the induced field depends on
  whether the semiconductor is p-type or n-type, so
  the thermoelectric effect can be used to
  determine the extent to which n- or p-type
  carriers dominate in a complex system.

34
Thermoelectric Effect

• When there is a temperature gradient in a
  thermoelectric material, an electric field
  appears.
• This happens in a pure metal since we can assume
  the system acts as a gas of free electrons.
• As in an ideal gas, the density of free electrons
  is greater at the colder end of the wire, and
  therefore the electrical potential should be
  higher at the warmer end and lower at the colder
  end.
• The free-electron model is not valid for
  semiconductors nevertheless, the conducting
  properties of a semiconductor are temperature
  dependent, as we have seen, and therefore it is
  reasonable to believe that semiconductors should
  exhibit a thermoelectric effect.
• This thermoelectric effect is sometimes called
  the Seebeck effect.

35
The Thomson and Peltier Effects

• In a normal conductor, heat is generated at the
  rate of I2R. But a temperature gradient across
  the conductor causes additional heat to be
  generated.
• This is the Thomson Effect.
• Here heat is generated if current flows toward
  the higher temperature and absorbed if toward the
  lower.
• The Peltier effect occurs when heat is generated
  at a junction between two conductors as current
  passes through the junction.

36
The Thermocouple

• An important application of the Seebeck
  thermoelectric effect is in thermometry. The
  thermoelectric power of a given conductor varies
  as a function of temperature, and the variation
  can be quite different for two different
  conductors.
• This difference makes possible the operation of
  a thermocouple.

37
11.3 Semiconductor Devices

• pn-junction Diodes
• Here p-type and n-type semiconductors are joined
  together.
• The principal characteristic of a pn-junction
  diode is that it allows current to flow easily in
  one direction but hardly at all in the other
  direction.
• We call these situations forward bias and
  reverse bias, respectively.

38
Operation of a pn-junction Diode
Figure 11.12 The operation of a pn-junction
diode. (a) This is the no-bias case. The small
thermal electron current (It) is offset by the
electron recombination current (Ir). The net
positive current (Inet) is zero. (b) With a DC
voltage applied as shown, the diode is in reverse
bias. Now Ir is slightly less than It. Thus there
is a small net flow of electrons from p to n and
positive current from n to p. (c) Here the diode
is in forward bias. Because current can readily
flow from p to n, Ir can be much greater than It.
Note In each case, It and Ir are electron
(negative) currents, but Inet indicates positive
current.
39
Bridge Rectifiers

• The diode is an important tool in many kinds of
  electrical circuits. As an example, consider the
  bridge rectifier circuit shown in Figure 11.14.
  The bridge rectifier is set up so that it allows
  current to flow in only one direction through the
  resistor R when an alternating current supply is
  placed across the bridge. The current through the
  resistor is then a rectified sine wave of the
  form
• This is the first step in changing alternating
  current to direct current. The design of a power
  supply can be completed by adding capacitors and
  resistors in appropriate proportions. This is an
  important application, because direct current is
  needed in many devices and the current that we
  get from our wall sockets is alternating current.
• Figure 11.14 Circuit diagram for a diode
  bridge rectifier.

(11.10)
40
Zener Diodes

• The Zener diode is made to operate under reverse
  bias once a sufficiently high voltage has been
  reached. The I-V curve of a Zener diode is shown
  in Figure 11.15. Notice that under reverse bias
  and low voltage the current assumes a low
  negative value, just as in a normal pn-junction
  diode. But when a sufficiently large reverse bias
  voltage is reached, the current increases at a
  very high rate.

Figure 11.16 A Zener diode reference circuit.
Figure 11.15 A typical I-V curve for a Zener
diode.
41
Light Emitting Diodes

• Another important kind of diode is the
  light-emitting diode (LED). Whenever an electron
  makes a transition from the conduction band to
  the valence band (effectively recombining the
  electron and hole) there is a release of energy
  in the form of a photon (Figure 11.17). In some
  materials the energy levels are spaced so that
  the photon is in the visible part of the
  spectrum. In that case, the continuous flow of
  current through the LED results in a continuous
  stream of nearly monochromatic light.

Figure 11.17 Schematic of an LED. A photon is
released as an electron falls from the conduction
band to the valence band. The band gap may be
large enough that the photon will be in the
visible portion of the spectrum.
42
Photovoltaic Cells

• An exciting application closely related to the
  LED is the solar cell, also known as the
  photovoltaic cell. Simply put, a solar cell takes
  incoming light energy and turns it into
  electrical energy. A good way to think of the
  solar cell is to consider the LED in reverse
  (Figure 11.18). A pn-junction diode can absorb a
  photon of solar radiation by having an electron
  make a transition from the valence band to the
  conduction band. In doing so, both a conducting
  electron and a hole have been created. If a
  circuit is connected to the pn junction, the
  holes and electrons will move so as to create an
  electric current, with positive current flowing
  from the p side to the n side. Even though the
  efficiency of most solar cells is low, their
  widespread use could potentially generate
  significant amounts of electricity. Remember that
  the solar constant (the energy per unit area of
  solar radiation reaching the Earth) is over 1400
  W/m2, and more than half of this makes it through
  the atmosphere to the Earths surface. There has
  been tremendous progress in recent years toward
  making solar cells more efficient.

Figure 11.18 (a) Schematic of a photovoltaic
cell. Note the similarity to Figure 11.17. (b) A
schematic showing more of the working parts of a
real photovoltaic cell. From H. M. Hubbard,
Science 244, 297-303 (21 April 1989).
43
Transistors

• Another use of semiconductor technology is in the
  fabrication of transistors, devices that amplify
  voltages or currents in many kinds of circuits.
  The first transistor was developed in 1948 by
  John Bardeen, William Shockley, and Walter
  Brattain (Nobel Prize, 1956). As an example we
  consider an npn-junction transistor, which
  consists of a thin layer of p-type semiconductor
  sandwiched between two n-type semiconductors. The
  three terminals (one on each semiconducting
  material) are known as the collector, emitter,
  and base. A good way of thinking of the operation
  of the npn-junction transistor is to think of two
  pn-junction diodes back to back.

Figure 11.22 (a) In the npn transistor, the base
is a p-type material, and the emitter and
collector are n-type. (b) The two-diode model of
the npn transistor. (c) The npn transistor symbol
used in circuit diagrams. (d) The pnp transistor
symbol used in circuit diagrams.
44
Transistors

• Consider now the npn junction in the circuit
  shown in Figure 11.23a. If the emitter is more
  heavily doped than the base, then there is a
  heavy flow of electrons from left to right into
  the base. The base is made thin enough so that
  virtually all of those electrons can pass through
  the collector and into the output portion of the
  circuit. As a result the output current is a very
  high fraction of the input current. The key now
  is to look at the input and output voltages.
  Because the base-collector combination is
  essentially a diode connected in reverse bias,
  the voltage on the output side can be made higher
  than the voltage on the input side. Recall that
  the output and input currents are comparable, so
  the resulting output power (current voltage) is
  much higher than the input power.

Figure 11.23 (a) The npn transistor in a voltage
amplifier circuit. (b) The circuit has been
modified to put the input between base and
ground, thus making a current amplifier. (c) The
same circuit as in (b) using the transistor
circuit symbol.
45
Field Effect Transistors (FET)

• The three terminals of the FET are known as the
  drain, source, and gate, and these correspond to
  the collector, emitter, and base, respectively,
  of a bipolar transistor.

Figure 11.25 (a) A schematic of a FET. The two
gate regions are connected internally. (b) The
circuit symbol for the FET, assuming the
source-to-drain channel is of n-type material and
the gate is p-type. If the channel is p-type and
the gate n-type, then the arrow is reversed. (c)
An amplifier circuit containing a FET.
46
Schottky Barriers

• Here a direct contact is made between a metal and
  a semiconductor. If the semiconductor is n-type,
  electrons from it tend to migrate into the metal,
  leaving a depleted region within the
  semiconductor.
• This will happen as long as the work function of
  the metal is higher (or lower, in the case of a
  p-type semiconductor) than that of the
  semiconductor.
• The width of the depleted region depends on the
  properties of the particular metal and
  semiconductor being used, but it is typically on
  the order of microns. The I-V characteristics of
  the Schottky barrier are similar to those of the
  pn-junction diode. When a p-type semiconductor is
  used, the behavior is similar but the depletion
  region has a deficit of holes.

47
Schottky Barriers
Figure 11.26 (a) Schematic drawing of a typical
Schottky-barrier FET. (b) Gain versus frequency
for two different substrate materials, Si and
GaAs. From D. A. Fraser, Physics of Semiconductor
Devices, Oxford Clarendon Press (1979).
48
Semiconductor Lasers

• Like the gas lasers described in Section 10.2,
  semiconductor lasers operate using population
  inversionan artificially high number of
  electrons in excited states
• In a semiconductor laser, the band gap determines
  the energy difference between the excited state
  and the ground state
• Semiconductor lasers use injection pumping, where
  a large forward current is passed through a diode
  creating electron-hole pairs, with electrons in
  the conduction band and holes in the valence
  band.
• A photon is emitted when an electron falls back
  to the valence band to recombine with the hole.

49
Semiconductor Lasers

• Since their development, semiconductor lasers
  have been used in a number of applications, most
  notably in fiber-optics communication.
• One advantage of using semiconductor lasers in
  this application is their small size with
  dimensions typically on the order of 10-4 m.
• Being solid-state devices, they are more robust
  than gas-filled tubes.

50
Integrated Circuits

• The most important use of all these semiconductor
  devices today is not in discrete components, but
  rather in integrated circuits called chips.
• Some integrated circuits contain a million or
  more components such as resistors, capacitors,
  and transistors.
• Two benefits miniaturization and processing
  speed.

51
Moores Law and Computing Power
Figure 11.29 Moores law, showing the progress
in computing power over a 30-year span,
illustrated here with Intel chip names. The
Pentium 4 contains over 50 million transistors.
Courtesy of Intel Corporation. Graph from
http//www.intel.com/research/silicon/mooreslaw.ht
m.
52
11.4 Nanotechnology

• Nanotechnology is generally defined as the
  scientific study and manufacture of materials on
  a submicron scale.
• These scales range from single atoms (on the
  order of .1 nm up to 1 micron (1000 nm).
• This technology has applications in engineering,
  chemistry, and the life sciences and, as such, is
  interdisciplinary.

53
Carbon Nanotubes

• In 1991, following the discovery of C60
  buckminsterfullerenes, or buckyballs, Japanese
  physicist Sumio Iijima discovered a new geometric
  arrangement of pure carbon into large molecules.
• In this arrangement, known as a carbon nanotube,
  hexagonal arrays of carbon atoms lie along a
  cylindrical tube instead of a spherical ball.

54
Structure of a Carbon Nanotube
Figure 11.30 Model of a carbon nanotube,
illustrating the hexagonal carbon pattern
superimposed on a tubelike structure. There is
virtually no limit to the length of the tube.
From http//www.hpc.susx.ac.uk/ewels/img/science
/nanotubes/.
55
Carbon Nanotubes

• The basic structure shown in Figure 11.30 leads
  to two types of nanotubes. A single-walled
  nanotube has just the single shell of hexagons as
  shown.
• In a multi-walled nanotube, multiple layers are
  nested like the rings in a tree trunk.
• Single-walled nanotubes tend to have fewer
  defects, and they are therefore stronger
  structurally but they are also more expensive and
  difficult to make.

56
Applications of Nanotubes

• By their strength they are used as structural
  reinforcements in the manufacture of composite
  materials
• (batteries in cell-phones use nanotubes in this
  way)
• Nanotubes have very high electrical and thermal
  conductivities, and as such lead to high current
  densities in high-temperature superconductors.

57
Nanoscale Electronics

• One problem in the development of truly
  small-scale electronic devices is that the
  connecting wires in any circuit need to be as
  small as possible, so that they do not overwhelm
  the nanoscale components they connect.
• In addition to the nanotubes already described,
  semiconductor wires (for example indium
  phosphide) have been fabricated with diameters as
  small as 5 nm.

58
Nanoscale Electronics

• These nanowires have been shown useful in
  connecting nanoscale transistors and memory
  circuits.
• These are referred to as nanotransistors.

59
Nanotechnology and the Life Sciences

• The complex molecules needed for the variety of
  life on Earth are themselves examples of
  nanoscale design.
• Examples of unusual materials designed for
  specific purposes include the molecules that make
  up claws, feathers, and even tooth enamel.

60
Information Science

• Its possible that current photolithographic
  techniques for making computer chips could be
  extended into the hard-UV or soft x-ray range,
  with wavelengths on the order of 1 nm, to
  fabricate silicon-based chips on that scale.
• Possible quantum effects as devices become
  smaller, specifically the superposition of
  quantum states possibly leading to quantum
  computing.